Equivalence of definitions of Frattini subgroup
This article gives a proof/explanation of the equivalence of multiple definitions for the term Frattini subgroup
View a complete list of pages giving proofs of equivalence of definitions
The definitions that we have to prove as equivalent
Definition in terms of maximal subgroups
For a group , the Frattini subgroup is defined as the intersection of all maximal subgroups of :
When there are no maximal subgroups, we define as the whole group
Definition as the set of nongenerators
For a group , the Frattini subgroup is defined as the set of all nongenerators. An element is termed a nongenerator if, whenever is such that generates , itself generates .
Facts used
Proof
The set of nongenerators is contained in every maximal subgroup
Given: A group , a nongenerator for , a maximal subgroup of
To prove:
Proof: Suppose (contradiction point) . Then, . Thus, the set is a generating set for . By the definition of nongenerator, we see that , as a set, is a generating set for . But this is a contradiction since the subgroup generated by is itself.
Thus, is forced.
Any element in the intersection of all maximal subgroups is a nongenerator
Given: A group , an element in the intersection of maximal subgroups of , a subset such that generates
To prove: generates
Proof: Let be the subgroup generated by . Suppose (contradiction point) is proper. Then, , since . Thus, is a 1-completed subgroup: it, along with one outside element, generates the whole group. By the above fact, there exists a maximal subgroup containing . But by assumption, is in every maximal subgroup, so , so , so , a contradiction.
Thus, the assumption that is proper is false, so , so generates .
References
Textbook references
- Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Exercise 25, Page 199 (Section 6.2)